The first issue I am looking into is to make sure that synchronous tag will fire on the secure page.
Second is that it seems that some Tealium tags fire on HTTP protocol on secure pages. How can I fix it?

Where can I find an information on https and http Tealium settings? How to make sure that everything is set up correctly including asynchronous and synchronous Tealium tags?

The utag.js, or utag.sync.js should be referenced in a URL beginning with // rather than http:// or https:// - that should ensure that the correct protocol for the utag files are used.
If you have a specific tag that is referencing HTTP on a secure page, it's probably worth raising a support case so that we can check the template.